This article is a draft and as such there might be typos and other inaccuracies.

In this article we’ll derive the matrix inversion lemma, also known as the Sherman-Morrisson-Woodbury formula. At first it might seem like a very boring piece of linear algebra, but it has a few nifty uses, as we’ll see in one of the followup articles.

Let’s start with the following block matrix:

$$
M = \begin{bmatrix}
A & U \\\
V & B
\end{bmatrix}
$$

We’ll do an LDU decomposition in two different ways, which basically direclty gives us the end formula. Eliminating the bototm left element we get the following: